On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
How to net a lot with little: small &egr;-nets for disks and halfspaces
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Reporting points in halfspaces
Computational Geometry: Theory and Applications
Fat Triangles Determine Linearly Many Holes
SIAM Journal on Computing
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
On linear-time deterministic algorithms for optimization problems in fixed dimension
Journal of Algorithms
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
Algorithms for Polytope Covering and Approximation
WADS '93 Proceedings of the Third Workshop on Algorithms and Data Structures
Covering Rectilinear Polygons with Axis-Parallel Rectangles
SIAM Journal on Computing
A constant-factor approximation algorithm for optimal terrain guarding
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Hitting sets when the VC-dimension is small
Information Processing Letters
Conflict-free colorings of shallow discs
Proceedings of the twenty-second annual symposium on Computational geometry
Improved Approximation Algorithms for Geometric Set Cover
Discrete & Computational Geometry
Proceedings of the twenty-fourth annual symposium on Computational geometry
Small-size ε-nets for axis-parallel rectangles and boxes
Proceedings of the forty-first annual ACM symposium on Theory of computing
Small-size ε-nets for axis-parallel rectangles and boxes
Proceedings of the forty-first annual ACM symposium on Theory of computing
On the set multi-cover problem in geometric settings
Proceedings of the twenty-fifth annual symposium on Computational geometry
Weighted geometric set cover via quasi-uniform sampling
Proceedings of the forty-second ACM symposium on Theory of computing
Small-Size $\eps$-Nets for Axis-Parallel Rectangles and Boxes
SIAM Journal on Computing
Tight lower bounds for the size of epsilon-nets
Proceedings of the twenty-seventh annual symposium on Computational geometry
Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Improved bound for the union of fat triangles
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
On the set multicover problem in geometric settings
ACM Transactions on Algorithms (TALG)
Weighted geometric set multi-cover via quasi-uniform sampling
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Small-size relative (p,ε)-approximations for well-behaved range spaces
Proceedings of the twenty-ninth annual symposium on Computational geometry
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We consider the following combinatorial problem: given a set of n objects (for example, disks in the plane, triangles), and an integer L ≥ 1, what is the size of the smallest subset of these n objects that covers all points that are in at least L of the objects? This is the classic question about the size of an L/n-net for these objects. It is well known that for fairly general classes of geometric objects the size of an L/n-net is O(n/L log n/L). There are some instances where this general bound can be improved, and this improvement is usually due to bounds on the combinatorial complexity (size) of the boundary of the union of these objects. Thus, the boundary of the union of m disks has size O(m), and this translates to an O(n/L) bound on the size of an L/n-net for disks. For m fat triangles, the size of the union boundary is O(m log log m), and this yields L/n-nets of size O(n/L log log n/L). Improved nets directly translate into an upper bound on the ratio between the optimal integral solution and the optimal fractional solution for the corresponding geometric set cover problem. Thus, for covering k points by disks, this ratio is O(1); and for covering k points by fat triangles, this ratio is O(log log k). This connection to approximation algorithms for geometric set cover is a major motivation for attempting to improve bounds on nets. Our main result is an argument that in some cases yields nets that are smaller than those previously obtained from the size of the union boundary. Thus for fat triangles, for instance, we obtain nets of size O(n/L log log log n). We use this to obtain a randomized polynomial time algorithm that gives an O(log log log k)-approximation for the problem of covering k points by the smallest subset of a given set of triangles.